Talk:PlanetPhysics/Vector Potential

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Let\, $\vec{U} = \vec{U}(x,\,y,\,z)$\, be a \htmladdnormallink{vector field}{http://planetphysics.us/encyclopedia/NeutrinoRestMass.html} in $\mathbb{R}^3$ with continuous partial derivatives.\, Then the following three conditions are equivalent: \begin{itemize} \item The surface integrals of $\vec{U}$ over all contractible closed surfaces $S$ vanish: $$\oint_S\vec{U}\cdot d\vec{S} = 0$$ \item The \htmladdnormallink{divergence}{http://planetphysics.us/encyclopedia/DivergenceOfAVectorField.html} of $\vec{U}$ vanishes everywhere in the \htmladdnormallink{field}{http://planetphysics.us/encyclopedia/VectorField.html}: $$\nabla\!\cdot\!\vec{U} = 0$$ \item There exists the {\em vector potential}\, $\vec{A} = \vec{A}(x,\,y,\,z)$\, of $\vec{U}$: $$\nabla\!\times\!\vec{A} = \vec{U}$$ \end{itemize}

Under those conditions, the vector field $\vec{U}$ is called \emph{solenoidal}.

\begin{thebibliography}{9} \bibitem{VV}{\sc K. V\"ais\"al\"a:} {\em Vektorianalyysi}. \,Werner S\"oderstr\"om Osakeyhti\"o, Helsinki (1961). \end{thebibliography}

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